Volume 29, Issue 6

We prove the Hölder continuity of a homeomorphism f defined on a bounded domain Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} with Lipschitz boundary. The homeomorphism f is assumed to belong to a certain Orlicz–Sobolev class and to satisfy a distortion condition near the boundary.

This article deals with the regularity of a solution for a non-homogeneous heat equation in Sobolev spaces. The problem is subject to Cauchy–Dirichlet boundary conditions considered in a rectangle or domain that can be transformed into a rectangle.

We give a transference maximum inequality establishing interrelation between the sign-convergence and the rearrangement convergence of functional series. We use this inequality to get a generalization of the Maurey and Pisier sign-permutation relationship and the general Garsia and Nikishin type local inequalities, as well as the Nikishin and Garsia convergence theorems on the almost sure convergence of functional series. We also get some maximum inequalities that might be useful in the analysis of the Kolmogorov Rearrangement Conjecture on the systems of convergence of orthogonal series as well as Garsia’s local form of the conjecture.

The Cauchy problem for an abstract hyperbolic equation with the Lipschitz continuous operator is considered in the Hilbert space. The operator corresponding to the elliptic part of the equation is the sum of operators A 1 , A 2 , … , A m {A_{1},A_{2},\ldots,A_{m}} . Each summand is a self-adjoint and positive definite operator. A parallel type decomposition scheme for an approximate solution of the stated problem is constructed. The main idea of the scheme is that on each local interval the classical difference problems are solved in parallel (independently from each other) with the operators A 1 , A 2 , … , A m {A_{1},A_{2},\ldots,A_{m}} . The weighted average of the obtained solutions is announced as an approximate solution at the right end of the local interval. The convergence of the proposed scheme is proved and the approximate solution error is estimated, as well as the error of the difference analogue for the first-order derivative for the case when the initial problem data satisfy the natural sufficient conditions for solution existence.

Regarding the question of how idempotent elements affect the reversible property of rings, we study a version of reversibility depending on idempotents. In this perspective, we introduce right (resp., left ) 𝑒-reversible rings . We show that this concept is not left-right symmetric. Basic properties of right 𝑒-reversibility in a ring are provided. Among others, it is proved that if 𝑅 is a semiprime ring, then 𝑅 is right 𝑒-reversible if and only if it is right 𝑒-reduced if and only if it is 𝑒-symmetric if and only if it is right 𝑒-semicommutative. Also, for a right 𝑒-reversible ring 𝑅, 𝑅 is a prime ring if and only if it is a domain. It is shown that the class of right 𝑒-reversible rings lies strictly between that of 𝑒-symmetric rings and right 𝑒-semicommutative rings. Some extensions of rings are studied in terms of 𝑒-reversibility.

In this paper, we study the multiplicity of weak solutions to the boundary value problem - Δ γ ⁢ u = f ⁢ ( x , u ) + g ⁢ ( x , u ) \displaystyle{-}\Delta_{\gamma}u=f(x,u)+g(x,u) in  ⁢ Ω , \displaystyle\phantom{}\text{in }\Omega, u = 0 \displaystyle u=0 on  ⁢ ∂ ⁡ Ω , \displaystyle\phantom{}\text{on }\partial\Omega, where Ω is a bounded domain with smooth boundary in ℝ N ( N ≥ 2 ) , {\mathbb{R}^{N}\ (N\geq 2),} f ⁢ ( x , ξ ) {f(x,\xi)} is odd in ξ, g ⁢ ( x , ξ ) {g(x,\xi)} is a perturbation term and Δ γ {\Delta_{\gamma}} is a subelliptic operator of the type Δ γ := ∑ j = 1 N ∂ x j ⁡ ( γ j 2 ⁢ ∂ x j ) , ∂ x j := ∂ ∂ ⁡ x j , γ := ( γ 1 , γ 2 , … , γ N ) . \Delta_{\gamma}:=\sum_{j=1}^{N}\partial_{x_{j}}(\gamma_{j}^{2}\partial_{x_{j}}% ),\qquad\partial_{x_{j}}:=\frac{\partial}{\partial x_{j}},\quad\gamma:=(\gamma% _{1},\gamma_{2},\dots,\gamma_{N}). By using the variant of Rabinowitz’s perturbation method, under some growth conditions on f and g , we show that there are infinitely many weak solutions to the problem.

In this paper, we consider a special approach to the investigation of a mixed boundary value problem (BVP) for the Laplace equation in the case of a three-dimensional bounded domain Ω ⊂ ℝ 3 {\Omega\subset\mathbb{R}^{3}} , when the boundary surface S = ∂ ⁡ Ω {S=\partial\Omega} is divided into two disjoint parts S D {S_{D}} and S N {S_{N}} where the Dirichlet—Neumann-type boundary conditions are prescribed, respectively. Our approach is based on the potential method. We look for a solution to the mixed boundary value problem in the form of a linear combination of the single layer and double layer potentials with the densities supported respectively on the Dirichlet and Neumann parts of the boundary. This approach reduces the mixed BVP under consideration to a system of pseudodifferential equations. The corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate L 2 {L_{2}} -based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the Sobolev space W 2 1 ⁢ ( Ω ) {W^{1}_{2}(\Omega)} . Using a special structure of the obtained pseudodifferential matrix operator, it is also shown that it is invertible in the L p {L_{p}} -based Besov spaces, which under appropriate boundary data implies C α {C^{\alpha}} -Hölder continuity of the solution to the mixed BVP in the closed domain Ω ¯ {\overline{\Omega}} with α = 1 2 - ε {\alpha=\frac{1}{2}-\varepsilon} , where ε > 0 {\varepsilon>0} is an arbitrarily small number.

In this article, we introduce a new kind of split monotone variational inclusion problem involving Cayley operator in the setting of infinite-dimensional Hilbert spaces. We develop a general iterative method to approximate the solution of the split monotone variational inclusion problem involving Cayley operator. Under some suitable conditions, a convergence theorem for the sequences generated by the proposed iterative scheme is established, which also solves certain variational inequality problems related to strongly positive linear operators. Finally, a numerical example is presented to study the efficiency of the proposed algorithm through MATLAB programming.

The purpose of this paper is to study duals and approximate duals of von Neumann–Schatten p-frames. We present a necessary and sufficient condition for the existence of a dual for a von Neumann–Schatten p-frame. Also, von Neumann–Schatten q-Riesz bases are characterized and it is shown that they share many useful properties with their corresponding notions in Hilbert spaces. Especially, it is focused on the uniqueness of their duals, and some equivalent conditions under which a von Neumann–Schatten p-frame admits a unique dual are derived. Moreover, approximate duality for von Neumann–Schatten p-Bessel sequences is introduced and some of the important properties of approximate duals are obtained, mainly their stability under different kinds of perturbations is considered.

The paper deals with problems of convergence of the series of moduli of the difference of general Fourier coefficients of functions from some differentiable class. It is shown that the good differentiable properties of a function do not guarantee the convergence of the series of moduli of the difference of general Fourier coefficients of this function. We explain the bounded variation of a sequence of general Fourier coefficients of a function from some functional class. We also find the conditions on the functions of an orthonormal system (ONS) under which the sequence of Fourier coefficients of any function from the differentiable class is of bounded variation. It is established that the resulting conditions are best possible. Studied here is the particular character of a subsequence of ONS.

We fix a continuous linear operator A : H → M {A:H\rightarrow M} acting between the Hilbert spaces H and M that admits a singular value decomposition (SVD). We consider the following ill-posed problem: for an element f ∈ M {f\in M} , find u ∈ H {u\in H} such that A ⁢ u = f {Au=f} and a generalized solution in the sense of Moore–Penrose u is sought that satisfies the equation A * ⁢ A ⁢ u = A * ⁢ f {A^{*}Au=A^{*}f} . Moreover, we fix an integer n ∈ ℕ 0 = { 0 , 1 , 2 , … } {n\in\mathbb{N}_{0}=\{0,1,2,\dots\}} and transfer this equation to a special Hilbert space D ⁢ ( ( A * ⁢ A ) - n ) {D((A^{*}A)^{-n})} of n -orbits. For an approximate solution of this equation in the case of a nonadaptive information on the right-hand side f , a linear spline algorithm is constructed. The specificity of the considered norm is that the approximate solution is the truncated singular value decomposition (TSVD) and does not depend on n . In the case n = 0 {n=0} , the space D ⁢ ( ( A * ⁢ A ) - n ) {D((A^{*}A)^{-n})} coincides with H and we obtain the results for the latter space. In the limiting case of the Fréchet–Hilbert space of all orbits D ⁢ ( ( A * ⁢ A ) - ∞ ) {D((A^{*}A)^{-\infty})} , the equation A * ⁢ A ⁢ u = A * ⁢ f {A^{*}Au=A^{*}f} becomes well-posed and was considered in [D. Ugulava and D. Zarnadze, On a linear generalized central spline algorithm of computerized tomography, Proc. A. Razmadze Math. Inst. 168 2015, 129–148]. It is also noted that the space D ⁢ ( ( A * ⁢ A ) - ∞ ) {D((A^{*}A)^{-\infty})} is the projective limit of the sequence of Hilbert spaces { D ⁢ ( ( A * ⁢ A ) - n ) } {\{D((A^{*}A)^{-n})\}} . The application of the obtained results for the computerized tomography problem, i.e., for the inversion of the Radon transform ℜ {\mathfrak{R}} with the SVD of Louis [A. K. Louis, Orthogonal function series expansions and the null space of the Radon transform, SIAM J. Math. Anal. 15 1984, 3, 621–633] in the space D ( ( ℜ * ℜ ) - n ) ) {D((\mathfrak{R}^{*}\mathfrak{R})^{-n}))} is given.

The aim of this paper is to give some discrete and continuous versions of Barbashin type theorems for the uniform polynomial stability and instability of evolution families in Banach spaces, by using Banach sequence spaces in ℋ ⁢ ( ℕ ≥ 1 ) {\mathcal{H}(\mathbb{N}_{\geq 1})} and Banach function spaces in ℋ ⁢ ( ℝ ≥ 1 ) {\mathcal{H}(\mathbb{R}_{\geq 1})} , respectively. Variants for uniform polynomial stability and instability of some well-known results in the exponential stability theory and polynomial stability theory are obtained.